#2 Review of Linear Systems and Fourier Transforms1Systemsp1 ( x1 , y1 )Imaging SystemS{ p1 ( x1 , y1 )} = p 2 ( x2 , y 2 )S{ }A system accepts an input signal and produces an output signal. Mathematically, a system can be described using an operator S{ } that maps a set of input functions onto a set of output functions. For imaging systems, the inputs and outputs are generally two dimensional complex-valued functions.2Examples of linear and nonlinear systemsLinear System Multiply by 5S{ p( x1 , y1 ) + q ( x1 , y1 )} = 5 p ( x1 , y1 ) + 5q( x1 , y1 )Linear since the input signals interact independentlySquareNonlinear SystemS{ p ( x1 , y1 ) + q ( x1 , y1 )} = p 2 ( x1 , y1 ) + q 2 ( x1 , y1 ) + 2 p ( x1 , y1 )q ( x1 , y1 )Not linear since the input signals interact with one another in this 3 term.Linear systems satisfy superposition and scaling propertiesSuppose we have a signal that can be composed of a sum of “elementary” functions. Response to an individual elementary function: Response to an input signal composed of these scaled elementary functions (inputted at the same time into the system):S{ap ( x1 , y1 ) + bq ( x1 , y1 )} = aS{ p ( x1 , y1 )} + bS{q ( x1 , y1 )}where a, b are constants (can be complex-valued)4Properties of Linear SystemsThe system treats each of the elementary functions p(x1,y1) and q(x1,y1) independently.S{ap ( x1 , y1 ) + bq ( x1 , y1 )} = aS{ p ( x1 , y1 )} + bS{q ( x1 , y1 )}Notice that the output depends on p and q independently.5Fourier transform is linear?Here, we write a square wave as a sum of sine waves.6Shift TheoremThe Fourier transform of a shifted function, g(t-a)F {g (t a )} = eProof:∞ i 2πfaG( f )F {g (t a )} =∞∫ g (t a) exp(i 2πft )dt∞Change variables: u=t-aF {g (t a )} = exp(i 2πfa ) ∫ g (u ) exp(i 2πfu )du= exp(i 2πfa )G ( f )7∞The Fourier Transform of a sum of two functions F(ω) f(t)F {af (t ) + bg (t )} = aF { f (t )} + bF {g (t )}g(t) t G(ω)ωtωF(ω) + G(ω)Also, constants factor out.f(t)+g(t)tω8Fourier transform is linear.A signal can also be decomposed into a sum of sinusoids.F {g }F 1{G}Real spaceFourier TransformFrequency spaceF {g } = G ( f x , f y ) = ∫ ∫ g ( x, y ) exp j 2π ( f x x + f y y ) dx dy∞ ∞[]Inverse Fourier TransformF1{G} = g ( x, y ) = ∫ ∫∞ G ( f x , f y )exp[ j 2π ( f x x + f y y )] df x df y∞9Signal Decomposition Using SinusoidsInverse Fourier TransformF1{G} = g ( x, y ) = ∫ ∫∞ G( f X , fY ) exp[ j 2π ( f X x + fY y )] df X∞dfYsignalweighting function for spatial frequencies fx and fyElementary function with spatial frequencies fx and fy Physically, we can think of this as elementary functions with different angles and different spatial periods10Similarity Theorem ExamplesFrequencyDomainWide in fx,Narrow in fyNarrow in fx,Wide in fyAmplitude PhaseCircular aperture Airy functionA function with circular symmetry can be described by the radialθg=,(g)ryf 1xf 1θL+ +。
